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Unformatted text preview: PROBABILITY AND DECISION MAKING 45-730 QUIZ 1 (OFFLINE) SOLUTIONS Problem 1 ( Sample Spaces and Set Theory ) . Consider the following random experiment : a (six-sided) die is thrown and a coin is flipped simultaneously. A basic outcome of this experiment is the result of both the die roll and the coin flip considered together. (1.1) How many basic outcomes are in the sample space? Solution. The basic outcomes of this experiment are listed as pairs ( D,C ), where the first coordinate represents the outcome of the die roll ( { 1 , 2 , 3 , 4 , 5 , 6 } ) and the second coordinate represents the outcome of the coin flip (Hhead, Ttail). { (1 ,H ) , (2 ,H ) , (3 ,H ) , (4 ,H ) , (5 ,H ) , (6 ,H ) , (1 ,T ) , (2 ,T ) , (3 ,T ) , (4 ,T ) , (5 ,T ) , (6 ,T ) } . where the first coordinate represents the outcome of the die roll and the second coordinate represents the outcome of the coin flip (Hhead, Ttail). There are 12 basic outcomes in the sample space. A faster way to count is to note that there are six choices for the outcome of the die roll and two choices for the outcome of the coin flip. This yields twelve different choices for both the die roll and the coin flip when considered together. (1.2) Let B be the event that an even number was rolled. How many basic outcomes are in B ? Solution. The subset B consists of the following basic outcomes: { (2 ,H ) , (4 ,H ) , (6 ,H ) , (2 ,T ) , (4 ,T )(6 ,T ) } . Thus there are six basic outcomes. One way to count is to realize that exactly half the die rolls will result in an even number and that the result of the coin flip is irrelevant. This means that exactly half of the total number of basic outcomes belong to B . A second way to count is as follows: there are exactly three choices for the die roll ( { 2 , 4 , 6 } ) and two choices for the coin flip ( { H,T } ) yielding six basic outcomes in B . (1.3) Let H be the event that the coin flip resulted in a head. How many basic outcomes are in H ? Solution. The subset H consists of the following basic outcomes: { (1 ,H ) , (2 ,H ) , (3 ,H ) , (4 ,H ) , (5 ,H )(6 ,H ) } . Thus there are six events in H . One way to count is to realize that exactly half the coin flips will result in a head and that the result of the die roll is irrelevant. This means that exactly half of the total number of basic outcomes belong to H . A second way to count is as follows: there are exactly six choices for the die roll ( { 1 , 2 , 3 , 4 , 5 , 6 } ) and only one choice for the coin flip ( { H } ) yielding six basic outcomes in H . (1.4) How many basic outcomes are in the union B H ? Solution. A basic outcome belong to the union BH if the basic outcome belongs to B or belongs to H or belongs to both B and H (the mathematical meaning of or covers the third(both) case and so explicitly listing the third(both) case is redundant). Thus the basic outcomes in B H are { (1 ,H ) , (2 ,H...
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